For a positive integer $m$ and a real number $c$, let $R = R(m,c,2)$ denote the discrete 2-color Rado number for the equation $x_1 + x_2 + \dots + x_m + c = 2x_0$. In other words, $R$ is the smallest integer such that for any coloring of the integers ${ 1, 2, \dots, R}$, there exist numbers $x_1, x_2, \dots, x_m, x_0$, all with the same color, such that $x_1 + x_2 + \dots + x_m + c = 2x_0$. In this article we show that if $m \geq 2$ and $c > 0$, then $$ \begin{ array}{ cc} R(m,c,2) = \begin{ cases} \infty & \text{ for $m$ even, $c$ odd} \newline \big\lceil \frac{ m}{ 2} \big\lceil \frac{ m+c}{ 2} \big\rceil + \frac{ c}{ 2} \big\rceil & \text{ otherwise.} \end{ cases} \end{ array} $$ For real numbers $a$ and $c$, we look at the 2-color Rado number for the equation $x_1 + c = ax_0$. We show that if $a > 1$ and $c > 0$, then the 2-color continuous Rado number is $$ R_\mathbb{ R}(1, c, a) = \begin{ cases} \infty & \text{ if $a=1$ } \newline \frac{ c}{ a-1} & \text{ otherwise.} \end{ cases} $$ From this, we will show that the discrete Rado number is $$ R(1, c, a) = \begin{ cases} \frac{ c}{ a-1} & \text{ if $ \left( a-1 \right) \mid c$} \newline \infty & \text{ otherwise.} \end{ cases} $$
Categories: math.CO